Group #10693899 — subgroups

Subgroups — #10693899

Order 768.

290 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#1567	C₂	1	1	yes	maximal, characteristic
2	#70520	C₂	1	2	yes	maximal
3	C₂×#4457	C₂	1	1	yes	maximal, characteristic
4	#69725	C₂	1	1	yes	maximal, characteristic
5	#67714	C₂	1	2	yes	maximal
6	#9626	—	3	1	no	maximal, Sylow
7	C₃×#423	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#822	—	2	2	no	—
10	#4925	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
12	C₄×#208	C₂²	1	1	yes	characteristic
13	C₃×#335	—	2	1	no	—
14	C₃×#335	—	2	1	no	—
15	#4558	—	2	2	no	—
16	#4457	C₄	1	2	yes	—
17	#4457	C₄	1	2	yes	—
18	#4455	—	2	1	no	—
19	#4454	—	2	1	no	—
20	#1950	—	3	2	no	—
21	#1567	D₃	1	1	yes	characteristic
22	C₂×#334	—	3	1	no	—
23	#1451	—	3	1	no	—
24	#1301	—	3	2	no	—
25	C₂×C₃×M₁₆	—	2	2	no	—
26	C₂²×C₃×C₈	Q₈	1	1	yes	characteristic
27	C₂²×C₃×C₈	Q₈	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
29	C₂×#208	D₄	1	1	yes	characteristic
30	C₂×#208	D₄	1	1	yes	characteristic
31	C₂²×C₃⋊C₈	—	2	2	no	—
32	C₃×C₄⋊C₈	—	2	2	no	—
33	C₃×C₄⋊C₈	—	2	2	no	—
34	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄×C₈	D₄	1	1	yes	characteristic
37	C₃×C₄×C₈	D₄	1	1	yes	characteristic
38	#830	Q₈	1	2	yes	—
39	#822	—	4	4	no	—
40	C₄⋊(C₃⋊C₈)	—	2	2	no	—
41	C₄⋊(C₃⋊C₈)	—	2	1	no	—
42	C₄⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
43	#803	—	2	1	no	—
44	C₄×C₃⋊C₈	—	2	1	no	—
45	C₄×C₃⋊C₈	—	2	1	no	—
46	#431	—	3	1	no	—
47	C₂×#108	—	6	2	no	—
48	#423	C₂×D₃	1	2	yes	—
49	C₂×C₄⋊C₈	—	3	2	no	—
50	C₄×M₁₆	—	3	1	no	—
51	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
52	#341	—	6	2	no	—
53	#335	—	2	1	no	—
54	#335	—	2	1	no	—
55	#334	—	3	2	no	—
56	#334	—	3	2	no	—
57	#322	—	6	1	no	—
58	#322	—	6	1	no	—
59	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
62	C₃×M₁₆	—	4	2	no	—
63	C₃×M₁₆	—	4	2	no	—
64	C₂×C₃×C₈	—	2	2	no	—
65	C₂×C₃×C₈	—	4	1	no	—
66	C₂×C₃×C₈	—	4	2	no	—
67	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
71	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
72	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
73	C₃×C₄²	#41	1	1	yes	characteristic
74	C₃×C₄²	#41	1	1	yes	characteristic
75	#208	—	2	2	no	—
76	#208	—	2	2	no	—
77	C₂×C₃⋊C₈	—	2	4	no	—
78	C₂×C₃⋊C₈	—	2	1	no	—
79	C₂×C₃⋊C₈	—	2	1	no	—
80	C₂×C₃⋊C₈	—	2	2	no	—
81	C₂×C₃⋊C₈	—	4	2	no	—
82	C₂×C₃⋊C₈	—	4	2	no	—
83	C₂×M₁₆	—	2	2	no	—
84	C₂×M₁₆	—	3	1	no	—
85	C₂×M₁₆	—	3	1	no	—
86	C₂²×C₈	—	6	2	no	—
87	C₂²×C₈	Dic₆	1	1	yes	characteristic
88	C₂²×C₈	Dic₆	1	1	yes	characteristic
89	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
90	#108	—	12	4	no	—
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	—	2	2	no	—
93	C₄⋊C₈	—	3	4	no	—
94	C₄⋊C₈	—	6	2	no	—
95	C₄⋊C₈	—	6	1	no	—
96	C₂²⋊C₈	—	3	2	no	—
97	#97	—	6	1	no	—
98	C₄×C₈	—	6	1	no	—
99	C₄×C₈	—	6	1	no	—
100	C₄×C₈	C₄×D₃	1	1	yes	characteristic
101	C₄×C₈	C₄×D₃	1	1	yes	characteristic
102	C₄×C₈	#67	1	1	yes	characteristic
103	C₄×C₈	#67	1	1	yes	characteristic
104	C₂³×C₃	C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	1	no	—
106	C₂×C₃×C₄	—	2	1	no	—
107	C₂×C₃×C₄	—	4	1	no	—
108	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
110	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
111	C₂×C₃×C₄	#121	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₃×C₄	#125	1	1	yes	characteristic
114	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
115	C₂×C₃×C₄	#137	1	2	yes	—
116	C₃×C₈	—	2	1	no	—
117	C₃×C₈	—	2	1	no	—
118	C₃×C₈	—	2	1	no	—
119	C₃×C₈	—	8	2	no	—
120	C₃×C₈	#108	1	2	yes	complemented
121	C₃⋊C₈	—	4	2	no	—
122	C₃⋊C₈	—	4	4	no	—
123	C₃⋊C₈	—	4	1	no	—
124	C₃⋊C₈	—	4	1	no	—
125	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
126	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
127	C₂²×C₄	D₃×Q₈	1	1	yes	characteristic
128	M₁₆	—	4	2	no	—
129	M₁₆	—	4	2	no	—
130	M₁₆	—	6	2	no	—
131	M₁₆	—	6	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	6	4	no	—
136	C₂×C₈	—	6	1	no	—
137	C₂×C₈	—	6	1	no	—
138	C₂×C₈	—	6	2	no	—
139	C₂×C₈	—	12	2	no	—
140	C₂×C₈	—	12	2	no	—
141	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
142	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
143	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
144	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
145	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
146	C₄²	#237	1	1	yes	characteristic
147	C₄²	#239	1	1	yes	characteristic
148	C₄²	C₂×#67	1	1	yes	characteristic
149	C₂²×C₃	—	2	1	no	—
150	C₂²×C₃	—	2	1	no	—
151	C₂²×C₃	#389	1	1	yes	characteristic
152	C₂²×C₃	#501	1	2	yes	—
153	C₃×C₄	—	2	1	no	—
154	C₃×C₄	—	2	1	no	—
155	C₃×C₄	—	2	1	no	—
156	C₃×C₄	C₂×#108	1	1	yes	characteristic
157	C₃×C₄	#430	1	1	yes	characteristic
158	C₃×C₄	#440	1	2	yes	—
159	C₃×C₄	#469	1	1	yes	characteristic
160	C₃×C₄	#470	1	1	yes	characteristic
161	C₂³	#888	1	1	yes	characteristic
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	#887	1	1	yes	characteristic
166	C₂×C₄	#890	1	1	yes	characteristic
167	C₂×C₄	#893	1	1	yes	characteristic
168	C₂×C₄	#911	1	2	yes	—
169	C₂×C₄	#918	1	2	yes	—
170	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
171	C₂×C₄	#938	1	1	yes	characteristic
172	C₂×C₄	#946	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	8	2	no	—
177	C₈	—	12	2	no	—
178	C₈	—	12	4	no	—
179	C₈	—	12	1	no	—
180	C₈	—	12	1	no	—
181	C₈	#822	1	2	yes	complemented
182	C₂×C₃	—	2	2	no	—
183	C₂×C₃	#1944	1	1	yes	characteristic
184	C₂×C₃	#1950	1	1	yes	characteristic
185	C₂×C₃	#1953	1	1	yes	characteristic
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	#4876	1	2	yes	—
189	C₂²	#4972	1	1	yes	characteristic
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	#4889	1	2	yes	—
194	C₄	C₂×#822	1	1	yes	characteristic
195	C₄	#5124	1	1	yes	characteristic
196	C₄	#5159	1	1	yes	characteristic
197	C₄	#5189	1	1	yes	characteristic
198	C₃	#9626	1	1	yes	characteristic, complemented, Sylow
199	C₂	—	2	2	no	—
200	C₂	#70520	1	1	yes	characteristic
201	C₂	#70614	1	1	yes	characteristic
202	C₂	#70706	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#1567

C₂

yes

maximal, characteristic

#70520

C₂

yes

maximal

C₂×#4457

C₂

yes

maximal, characteristic

#69725

C₂

yes

maximal, characteristic

#67714

C₂

yes

—

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₂×#822

—

#4925

C₂²

yes

characteristic

C₂×C₄⋊(C₃⋊C₈)

C₂²

yes

—

C₄×#208

C₂²

yes

—

—

—

C₄

yes

—

#4457

C₄

yes

—

—

—

—

D₃

yes

—

—

—

C₂×C₃×M₁₆

—

C₂²×C₃×C₈

Q₈

yes

characteristic

C₂²×C₃×C₈

Q₈

yes

characteristic

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×#208

D₄

yes

characteristic

C₂×#208

D₄

yes

characteristic

C₂²×C₃⋊C₈

—

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

—

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

characteristic

#830

Q₈

yes

—

#822

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

C₂×C₄

yes

—

#803

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

#431

—

C₂×#108

—

#423

C₂×D₃

yes

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₃

yes

—

—

—

—

—

—

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×C₃×C₄

C₂×Q₈

yes

characteristic

C₂²×C₃×C₄

C₂×Q₈

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄⋊C₄

yes

characteristic

C₂×C₃×C₈

C₄⋊C₄

yes

characteristic

C₂×C₃×C₈

C₄⋊C₄

yes

characteristic

C₂×C₃×C₈

C₄⋊C₄

yes

characteristic

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

C₃×C₄²

yes

C₃×C₄²

yes

—

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

Dic₆

yes

characteristic

C₂²×C₈

Dic₆

yes

characteristic

C₂×C₄²

C₂²×D₃

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₄×D₃

yes

characteristic

101

C₄×C₈

C₄×D₃

yes

characteristic

102

C₄×C₈

#67

yes

characteristic

103

C₄×C₈

#67

yes

characteristic

104

C₂³×C₃

C₄⋊Q₈

yes

characteristic

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

—

108

C₂×C₃×C₄

C₂×C₄⋊C₄

yes

characteristic

109

C₂×C₃×C₄

C₄×D₄

yes

characteristic

110

C₂×C₃×C₄

C₄×Q₈

yes

characteristic

111

C₂×C₃×C₄

#121

yes

characteristic

112

C₂×C₃×C₄

C₂²⋊Q₈

yes

characteristic

113

C₂×C₃×C₄

#125

yes

characteristic

114

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

115

C₂×C₃×C₄

#137

yes

—

116

C₃×C₈

—

117

C₃×C₈

—

118

C₃×C₈

—

119

C₃×C₈

—

120

C₃×C₈

#108

yes

complemented

121

C₃⋊C₈

—

122

C₃⋊C₈

—

123

C₃⋊C₈

—

124

C₃⋊C₈

—

125

C₂²×C₄

C₂×Dic₆

yes

characteristic

126

C₂²×C₄

D₃×D₄

yes

characteristic

127

C₂²×C₄

D₃×Q₈

yes

characteristic

128

M₁₆

—

129

M₁₆

—

130

M₁₆

—

131

M₁₆

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

Dic₃⋊C₄

yes

characteristic

142

C₂×C₈

Dic₃⋊C₄

yes

characteristic

143

C₂×C₈

Dic₃⋊C₄

yes

characteristic

144

C₂×C₈

Dic₃⋊C₄

yes

characteristic

145

C₄²

C₂×C₄×D₃

yes

characteristic

146

C₄²

#237

yes

characteristic

147

C₄²

#239

yes

characteristic

148

C₄²

C₂×#67

yes

characteristic

149

C₂²×C₃

—

150

C₂²×C₃

—

151

C₂²×C₃

#389

yes

characteristic

152

C₂²×C₃

#501

yes

—

153

C₃×C₄

—

154

C₃×C₄

—

155

C₃×C₄

—

156

C₃×C₄

C₂×#108

yes

characteristic

157

C₃×C₄

#430

yes

characteristic

158

C₃×C₄

#440

yes

—

159

C₃×C₄

#469

yes

characteristic

160

C₃×C₄

#470

yes

characteristic

161

C₂³

#888

yes

characteristic

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

#887

yes

characteristic

166

C₂×C₄

#890

yes

characteristic

167

C₂×C₄

#893

yes

characteristic

168

C₂×C₄

#911

yes

—

169

C₂×C₄

#918

yes

—

170

C₂×C₄

C₂×Dic₃⋊C₄

yes

characteristic

171

C₂×C₄

#938

yes

characteristic

172

C₂×C₄

#946

yes

characteristic

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

#822

yes

complemented

182

C₂×C₃

—

183

C₂×C₃

#1944

yes

characteristic

184

C₂×C₃

#1950

yes

characteristic

185

C₂×C₃

#1953

yes

characteristic

186

C₂²

—

187

C₂²

—

188

C₂²

#4876

yes

—

189

C₂²

#4972

yes

characteristic

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

#4889

yes

—

194

C₄

C₂×#822

yes

characteristic

195

C₄

#5124

yes

characteristic

196

C₄

#5159

yes

characteristic

197

C₄

#5189

yes

characteristic

198

C₃

#9626

yes

characteristic, complemented, Sylow

199

C₂

—

200

C₂

#70520

yes

characteristic

201

C₂

#70614

yes

characteristic

202

C₂

#70706

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#938
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1567	C₂
socle	C₂²×C₃	#389
prime radical	C₂³×C₃	C₄⋊Q₈
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1567	D₃
hypocenter	C₃	#9626
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

subgroup

G/H

center

C₂×C₄

#938

commutator

C₃×C₄²

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

C₂

C₂²×C₃

C₂³×C₃

C₄⋊Q₈

cosocle

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

N	H ≅ G/N	type
120. C₃×C₈	#108	semidirect
181. C₈	#822	semidirect
198. C₃	#9626	semidirect

H ≅ G/N

type

120. C₃×C₈

#108

semidirect

181. C₈

#822

semidirect

198. C₃

#9626

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#1567	2
#70520	2
C₂×#4457	2
#69725	2
#67714	2
#9626	3

subgroup

index

Subgroups — #10693899

Conjugacy classes of subgroups

Complements

Maximal subgroups